Spatial hierarchy analysis.

The manner in which the data are collected provides information about
aggregation of disease at different levels in a spatial hierarchy (Hughes et
al. 1997). For example, a sampling unit (upper level) can be reported as
"healthy", if no diseased leaves (lower level) were found within the sampling
unit.

spatial_hier(low, high)

Arguments

low

An list of intensity objects.

high

An list of intensity objects.

Details

In a pairwise comparison between levels, the probability that an individual
at the lower hierarchical level is diseased is denoted plow, and phigh refers
to the probability of disease at the higher level. The relationship between
these two probabilities can be written as

phigh = 1 - (1 - plow)^nu

where n is a parameter ranging from 0 to the corresponding number of
individuals at the hierarchical level referenced by plow. If the value of n
is equal to the number of individuals at the lower hierarchical level
contained in a unit of the higher level (n low ), this suggests that there is
no aggregation of disease incidence at the lower level. Conversely, a value
of n less than n low is indicative of aggregation at that level. The value of
n can be interpreted as an effective sample size (Hughes and Gottwald 1999;
Madden and Hughes 1999) in the statistical sense that its value indicates the
number of independent pieces of information at the lower level. Here, the
effective sample size concerns the equating of the zero-term of the binomial
distribution with the zero-term of an overdispersed distribution, as
described in Madden and Hughes (1999). Using the complementary log-log
transformation, CLL(x) = ln(-ln(1-x)), one can rewrite the Equation 5 as
follows (Madden et al. 2007):

CLL(phigh) = ln(nu) + CLL(plow)

from which the value of ln(n) can be obtained as the intercept of a linear
regression when the slope is constrained to 1.